1 Answer
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Every point $P = (a,b)$ on the square lies on some horizontal line segment $y = b$. If $a \neq 0$ -- i.e., if $P$ is not the leftmost point on its horizontal line segment -- then you can build a sequence of points $(x_n,b)$ to the left of $a$ on that line segment converging to $P$. If $a \neq 1$ -- i.e., if $P$ is not the rightmost point on its horizontal line segment -- then you can build a sequence of points $(x_n,b)$ to the right of $a$ on that line segment converging to $(a,b)$.

Obviously $P$ can't be both the leftmost point and the rightmost point, so the conclusion is...